Singular Matrix (Degenerate Matrix)?

Singular Matrix, also known as Degenerate Matrix, is a square matrix that does not have an inverse. Not having an inverse means that its determinant is zero.

A matrix is considered singular if its determinant is 0.

Example of a 2 x 2 singular matrix:

\(\begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix} \)

Here the determinant is 1 * 4 - 2 * 2 = 0. So, this matrix is singular.

What does it mean when a matrix is singular?

If A matrix is singular, then there does not exists a matrix B. Such that when A and B are multiplied, the result is the identity matrix.

A singular matrix cannot solve a system of linear equations. A singular matrix either gives no solutions or infinitely many solutions.

what is a good indicator if a matrix being singular?

A matrix is singular if the rows and columns of matrix is proportionally dependent. It means that one or more rows or columns of the matrix can be expressed as a linear combination of other columns or rows (without a constant term).

In Simple Words: If any rows(or columns) of a matrix is equal to its other columns(or rows) multiplied by a constant (weighted sum), the matrix is singular.

Its difficult to distinguish between a singular and nearly singular matrix. A good indicator is to check the determinant of the matrix. If the determinant is nearly zero, the matrix is nearly singular.

What is a singular correlation matrix?

Singular correlation matrix is a matrix which has at least one linearly dependent row or column. It means the determinant of the matrix is zero.

The correlation matrix is formed by calculating the correlation coefficient between pairs of variables.

If determinant of the correlation matrix is zero, there is a perfect linear relationship between two or more variables, also known as multicollinearity.

In case of multicollinearity, it is a good practice to remove of the variables from the statistical analysis to make the matrix non-singular.


Frequently Asked Questions

How to check if a matrix is singular?

To check if a matrix is singular, calculate the determinant of the matrix. If the determinant is zero, the matrix is singular.

Can a non-square matrix be singular matrix?

No, for a matrix to be singular, it must be a square matrix. Because only square matrices have determinants.

Can a singular matrix be positive definite ?

No, a singular matrix cannot be positive definite. A positive definite matrix must have all positive eigenvalues.

Example of 3 x 3 singular matrix?

\(\begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{bmatrix} \)

If you observe, the third row is a linear combination of first two rows. This is a linear dependent matrix.

So, the determinant of this matrix is zero. Hence, this is a singular matrix.

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